Orthogonal planes are two planes that intersect at a right angle (90 degrees). In three-dimensional space, a plane is defined as a flat, two-dimensional surface that extends infinitely in all directions within its dimension. The concept of orthogonality is central to many areas of mathematics and its applications, including geometry, linear algebra, and vector calculus.
Orthogonal planes are two planes that intersect at a right angle (90 degrees). In three-dimensional space, a plane is defined as a flat, two-dimensional surface that extends infinitely in all directions within its dimension. The concept of orthogonality is central to many areas of mathematics and its applications, including geometry, linear algebra, and vector calculus.
To understand orthogonal planes, it is important to first understand the concept of a normal vector. A normal vector to a plane is a vector that is perpendicular to every line lying on the plane. If two planes are orthogonal, their normal vectors are also orthogonal to each other.
Let's consider two planes in three-dimensional space:
Plane 1: \( Ax + By + Cz + D = 0 \)
Plane 2: \( A'x + B'y + C'z + D' = 0 \)
Here, \( A, B, C, D \) and \( A', B', C', D' \) are constants that define the respective planes. The vectors \( \mathbf{n} = (A, B, C) \) and \( \mathbf{n'} = (A', B', C') \) are the normal vectors to Plane 1 and Plane 2, respectively.
For the two planes to be orthogonal, their normal vectors must be orthogonal. This means that the dot product of the normal vectors must be zero:
\[ \mathbf{n} \cdot \mathbf{n'} = A \cdot A' + B \cdot B' + C \cdot C' = 0 \]
If this condition is satisfied, then Plane 1 and Plane 2 are orthogonal.
Let's go through a step-by-step process to determine if two given planes are orthogonal:
Step 1: Identify the normal vectors of the planes.
For Plane 1 with equation \( Ax + By + Cz + D = 0 \), the normal vector is \( \mathbf{n} = (A, B, C) \).
For Plane 2 with equation \( A'x + B'y + C'z + D' = 0 \), the normal vector is \( \mathbf{n'} = (A', B', C') \).
Step 2: Compute the dot product of the normal vectors.
Calculate the dot product \( \mathbf{n} \cdot \mathbf{n'} \) using the formula:
\[ \mathbf{n} \cdot \mathbf{n'} = A \cdot A' + B \cdot B' + C \cdot C' \]
Step 3: Check for orthogonality.
If the dot product is zero, \( \mathbf{n} \cdot \mathbf{n'} = 0 \), then the planes are orthogonal. If the dot product is not zero, the planes are not orthogonal.
Example:
Let's determine if the following two planes are orthogonal:
Plane 1: \( 2x - 3y + 4z + 5 = 0 \)
Plane 2: \( 6x + 9y - 8z + 10 = 0 \)
Step 1: Identify the normal vectors.
For Plane 1, \( \mathbf{n} = (2, -3, 4) \).
For Plane 2, \( \mathbf{n'} = (6, 9, -8) \).
Step 2: Compute the dot product.
\[ \mathbf{n} \cdot \mathbf{n'} = (2) \cdot (6) + (-3) \cdot (9) + (4) \cdot (-8) \]
\[ \mathbf{n} \cdot \mathbf{n'} = 12 - 27 - 32 \]
\[ \mathbf{n} \cdot \mathbf{n'} = -47 \]
Step 3: Check for orthogonality.
Since \( \mathbf{n} \cdot \mathbf{n'} \neq 0 \), the planes are not orthogonal.
In conclusion, the concept of orthogonal planes is based on the orthogonality of their normal vectors, and it can be determined by calculating the dot product of these normal vectors. If the result is zero, the planes are orthogonal; otherwise, they are not.